The motley of mathematics a study of Wittgenstein’s philosophy of mathematics

Abstract

In this thesis I try to examine Wittgenstein's philosophy of mathematics, both against the background of early twentieth century foundational studies and against the background of Wittgenstein’s general philosophical position. I have tried to explain Wittgenstein's objections to the foundational programmes and to show that they are consistent with and understandable in terms of, Wittgenstein's general philosophical outlook. In chapters one to four I discuss Wittgenstein's remarks on the mainstream foundational schools: logicism, intuitionism, formalism and strict finitism; and try to explain that in addition to the technical difficulties inherent in each, on Wittgenstein's view they are pointless endeavours, as mathematics has no need of foundations. In chapter five I discuss proof, and elaborate upon various points made in preceding chapters. In chapters six to eight I discuss the connections between the philosophy of mathematics, and Wittgenstein’s other central concerns (language games and forms of life, scepticism and rule following, and philosophy of mind) drawing parallels and trying to gain a fuller understanding of the former in terms of the latter. Chapter nine concludes my discussion, with an attempt to evaluate the significance of Wittgenstein's contribution; and to assess the principal objections levelled against it. Whilst not wholeheartedly embracing Wittgenstein’s position, I have frequently found myself defending him, and I do this because I think all too many of the criticisms of Wittgenstein derive from misunderstanding. The principal causes of misunderstanding are: (i) an over selective reading of Wittgenstein and a failure to understand individual remarks in their wider context. (ii) a failure to appreciate Wittgenstein's distinction between mathematics and philosophy; leading to a muddle between Wittgenstein's mathematical and his philosophical remarks, and much unfair criticism, I have particularly tried to establish Wittgenstein’s views on central questions, and to refute criticisms which derive from misunderstandings of these views.